Pieri’s formula for generalized Schur polynomials

Young's lattice, the lattice of all Young diagrams, has the Robinson-Schensted-Knuth correspondence, the correspondence between certain matrices and pairs of semi-standard Young tableaux with the same shape. Fomin introduced generalized Schur operators to generalize the Robinson-Schensted-Knuth correspondence. In this sense, generalized Schur operators are generalizations of semi-standard Young tableaux. We define a generalization of Schur polynomials as expansion coefficients of generalized Schur operators. We show that the commutation relation of generalized Schur operators implies Pieri's formula for generalized Schur polynomials.     

Recursions associated to trapezoid,symmetric and rotation symmetric functions over Galois fields

Rotation symmetric Boolean functions are invariant under circular translation of indices. These functions have very rich cryptographic properties and have been used in different cryptosystems. Recently, Thomas Cusick proved that exponential sums of rotation symmetric Boolean functions satisfy homogeneous linear recurrences with integer coefficients. In this work, a generalization of this result is proved over any Galois field. That is, exponential sums over Galois fields of some rotation symmetric polynomials satisfy linear recurrences with integer coefficients. In the particular case of ${\mathbb{F}}_2$, an elementary method is used to obtain explicit recurrences for exponential sums of some of these functions. The concept of trapezoid Boolean function is also introduced and it is showed that the linear recurrences that exponential sums of trapezoid Boolean functions satisfy are the same as the ones satisfied by exponential sums of the corresponding rotations symmetric Boolean functions. Finally, it is proved that exponential sums of trapezoid and symmetric polynomials also satisfy linear recurrences with integer coefficients over any Galois field ${\mathbb{F}}_q$. Moreover, the Discrete Fourier Transform matrix and some Complex Hadamard matrices appear as examples in some of our explicit formulas of these recurrences.     

两类对称函数的Schur凸性

本文用一种新方法研究两类对称函数的Schur凸性.首先,对x=(x1,...,xn)∈(-∞,1)n∪(1,+∞)n和r∈{1,2,...,n},讨论Guan(2007)定义的对称函数Fn(x,r)=Fn(x1,x2,...,xn;r)=∑1≤i1≤i2≤···≤ir≤n r∏j=1xij/(1-xij)的Schur凸性,其中i1,i2,...,in为正整数;推广褚玉明等人(2009)的主要结果,因而用新方法推广并解决Guan(2007)提出的一个公开问题.然后,对x=(x1,...,xn)∈(-∞,1)n∪(1,+∞)n和r∈{1,2,...,n},研究本文定义的对称函数Gn(x,r)=Gn(x1,x2,...,xn;r)=∑1≤i1≤i2≤···≤ir≤n(r∏j=1xij/(1-xij))1/r的Schur凸性、Schur乘性凸性和Schur调和凸性,其中i1,i2,...,in为正整数.作为应用,用Schur凸函数自变量的双射变换得到其他几类对称函数的Schur凸性,用控制理论建立一些不等式,特别地,由此给出Sharpiro不等式和Ky Fan不等式一个共同的推广,导出Safta猜想在高维空间的推广.     

Some families of generating functions for the extended Srivastava polynomials

Almost four decades ago, H.M. Srivastava considered a general family of univariate polynomials, the Srivastava polynomials, and initiated a systematic investigation for this family [10]. In 2001, B. González, J. Matera and H.M. Srivastava extended the Srivastava polynomials by inserting one more parameter [4]. In this study we obtain a family of linear generating functions for these extended polynomials. Some illustrative results including Jacobi, Laguerre and Bessel polynomials are also presented. Furthermore, mixed multilateral and multilinear generating functions are derived for these polynomials.     

New families of generating functions for certain class of three-variable polynomials

Recently, Srivastava, Özarslan and Kaanoglu have introduced certain families of three and two variable polynomials, which include Lagrange and Lagrange-Hermite polynomials, and obtained families of two-sided linear generating functions between these families [H.M. Srivastava, M.A. Özarslan, C. Kaanoglu, Some families of generating functions for a certain class of three-variable polynomials, Integr. Transform. Spec. Funct. iFirst (2010) 1-12]. The main object of this investigation is to obtain new two-sided linear generating functions between these families by applying certain hypergeometric transformations. Furthermore, more general families of bilinear, bilateral, multilateral finite series relationships and generating functions are presented for them.     

Schur convexity for a class of symmetric functions

The Schur convexity and concavity of a class of symmetric functions are discussed, and an open problem proposed by Guan in Some properties of a class of symmetric functions is answered. As consequences, some inequalities are established by use of the theory of majorization.     

Using Hook Schur Functions to Compute Matrix Cocharacters

We develop a new integration method based on hook Schur functions instead of Schur functions to compute the cocharacters of matrices. We then use this method to compute some of the multiplicities in the cocharacter sequence of 3 × 3 matrices.     

Green polynomials at roots of unity and Springer modules for the symmetric groups

We consider the Green polynomials at roots of unity. We obtain a recursive formula for the Green polynomials at roots of unity whose orders do not exceed some positive integer. The formula is described in a combinatorial manner. The coefficients of the recursive formula are realized by the cardinality of a set of permutations. The formula gives an interpretation of a combinatorial property on a family of graded modules for the symmetric group in terms of representation theory.     

q-Analogs of symmetric function operators

For any homomorphism V on the space of symmetric functions, we introduce an operation that creates a q-analog of V. By giving several examples we demonstrate that this quantization occurs naturally within the theory of symmetric functions. In particular, we show that the Hall-Littlewood symmetric functions are formed by taking this q-analog of the Schur symmetric functions and the Macdonald symmetric functions appear by taking the q-analog of the Hall-Littlewood symmetric functions in the parameter t. This relation is then used to derive recurrences on the Macdonald q,t-Kostka coefficients.RésuméPour un homomorphisme V sur l'espace des fonctions symétriques, nous présentons une opération qui crée un q-analogue de V. En donnant plusieurs exemples nous démontrons que cette quantization se produit naturellement dans la théorie de fonctions symétriques. En particulier, nous prouvons que les fonctions symétriques de Hall-Littlewood sont constituées en prenant ce q-analogue des fonctions symétriques de Schur et les fonctions symétriques de Macdonald apparaissent en prenant le q-analogue des fonctions symétriques de Hall-Littlewood dans le paramètre t. Cette relation est alors employée pour dériver des récurrence sur les coefficients Macdonald q,t-Kostka.     

Quasisymmetric and Schur expansions of cycle index polynomials

Given a subgroup $G$ of the symmetric group $S_n$, the cycle index polynomial ${\mathrm{cyc}}_G$ is the average of the power-sum symmetric polynomials indexed by the cycle types of permutations in $G$. By Pólya’s Theorem, the monomial expansion of ${\mathrm{cyc}}_G$ is the generating function for weighted colorings of $n$ objects, where we identify colorings related by one of the symmetries in $G$. This paper develops combinatorial formulas for the fundamental quasisymmetric expansions and Schur expansions of certain cycle index polynomials. We give explicit bijective proofs based on standardization algorithms applied to equivalence classes of colorings. Subgroups studied here include Young subgroups of $S_n$, the alternating groups $A_n$, direct products, conjugate subgroups, and certain cyclic subgroups of $S_n$ generated by $(1,2,\dots ,k)$. The analysis of these cyclic subgroups when $k$ is prime reveals an unexpected connection to perfect matchings on a hypercube with certain vertices identified.     

Some generating functions of Jacobi polynomials

In this paper, Weisner’s group-theoretic method of obtaining generating functions is utilized in the study of Jacobi polynomialsP> _n ^(a,ß)(x) by giving suitable interpretations to the index (n) and the parameter (β) to find out the elements for constructing a six-dimensional Lie algebra.     

Some families of generating functions for a class of bivariate polynomials

The main purpose of this paper is to present various families of generating functions for a class of polynomials in two variables. Furthermore, several general classes of bilinear, bilateral or mixed multilateral generating functions are obtained for these polynomials.     

一类对称函数的Schur凸性及其应用

对x=(x_1,…,x_n)∈[0,1)~n∪(1,+∞o)~n,定义对称函数■其中r∈N,i_1,i_2,…,i_n为非负整数.研究了F_n(x,r)的Schur凸性、Schur乘性凸性和Schur调和凸性.作为应用,用控制理论建立了一些不等式,特别地,给出了高维空间的一些新的几何不等式.     

The Schur algorithm and coefficient characterizations for generalized Schur functions

In this paper we analyze the existence of a Schur algorithm and obtain coefficient characterizations for the functions in a generalized Schur class. An application to an interpolation problem of Carathéodory type raised by M.G. Krein and H. Langer is indicated.

     

Quasisymmetric Schur functions

We introduce a new basis for quasisymmetric functions, which arise from a specialization of nonsymmetric Macdonald polynomials to standard bases, also known as Demazure atoms. Our new basis is called the basis of quasisymmetric Schur functions, since the basis elements refine Schur functions in a natural way. We derive expansions for quasisymmetric Schur functions in terms of monomial and fundamental quasisymmetric functions, which give rise to quasisymmetric refinements of Kostka numbers and standard (reverse) tableaux. From here we derive a Pieri rule for quasisymmetric Schur functions that naturally refines the Pieri rule for Schur functions. After surveying combinatorial formulas for Macdonald polynomials, including an expansion of Macdonald polynomials into fundamental quasisymmetric functions, we show how some of our results can be extended to include the t parameter from Hall-Littlewood theory.     

Some properties of a class of symmetric functions and its applications

For , the symmetric functions are defined by where , and are non‐negative integers. In this paper, the Schur convexity, geometric Schur convexity and harmonic Schur convexity of are investigated. As applications, Schur convexity for the other symmetric functions is obtained by a bijective transformation of independent variable for a Schur convex function, some analytic and geometric inequalities are established by using the theory of majorization, in particular, we derive from our results a generalization of Sharpiro's inequality, and give a new generalization of Safta's conjecture in the n‐dimensional space and others.